Nash Equilibrium
Consider the battle of the sexes
Man\Woman opera ballet
opera 1,4 0,0
ballet 0,0 4,1
In this game, there is no dominant strategy. But suppose W is playing opera. Then,
the best thing M can do is to play opera, too. Thus opera is a best-response for M
against opera. Similarly, opera is a best-response for Wagainst opera. Thus, at (opera,
opera), neither party wants to take a different action. This is a Nash Equilibrium.
More formally:
Definition 12 For any player i, a strategy s
BR
i
is a best response to s
−i
if and only if
u
i
(s
BR
i
, s
−i
) ≥ u
i
(s
i
, s
−i
),
∀
s
i
∈
S
i
This definition is identical to that of a dominant strategy except that it is not for
all s
−i
∈
S
−i
but for a specific strategy s
−i
. If it were true for all s
−i
, then S
BR
i
would
also be a dominant strategy, which is a stronger requirement than being a best response
against some strategy s
−i
.
Definition 13 A strategy profile (s
NE
1
, .
..s
NE
N
) is a Nash Equilibrium if and only if s
NE
i
is a best-response to s
NE
−i
= (s
NE
1
, .
..s
NE
i−1
, s
NE
i+1
, .
..s
NE
N
) for each i. That is, for all i, we
have that
u
i
(s
NE
i
, s
NE
−i